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Repdigits in generalized Pell sequences. (English) Zbl 07285963
Summary: For an integer $$k\ge 2$$, let $$(P_n^{(k)})_n$$ be the $$k$$-generalized Pell sequence which starts with $$0,\ldots ,0,1$$ ($$k$$ terms) and each term afterwards is given by the linear recurrence $$P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)}$$. In this paper, we find all $$k$$-generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence $$(P_n^{(2)})_n$$.

MSC:
 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11J86 Linear forms in logarithms; Baker’s method
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References:
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